 # 10. Prove the following generalization of Proposition 2.20. Let G a...

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10. Prove the following generalization of Proposition 2.20. Let G and 2 be open in C and suppose f and h are functions defined on G, g : 2 C and suppose that f (G) C S. Suppose that g and h are analytic, 8' (w) + 0 for any w, that f is continuous, h is one-one, and that they satisfy h(z) = 8(f(2)) for Z in G. Show that f is analytic. Give a formula for f'(z). 11. Suppose that f: G C is a branch of the logarithm and that n is an integer. Prove that z" = exp (nf(z)) for all Z in G. 12. Show that the real part of the function 2/2 is always positive. 13. Let G = C - { Z € R : Z V 0} and let n be a positive integer. Find all analytic functions f: G C such that Z = (f(z))" for all Z € G. 14. Suppose f: G C is analytic and that G is connected. Show that if f (z) is real for all Z in G then f is constant. { 1 15. For r > 0 let A = w : w = exp - where 0 < 1/11 < r ; determine the Z set A. G maximal? Are f and g analytic? 17. Give the principal branch of V 1 - z. 19. Let G be a region and define G* = If f: G C is analytic prove that f * : G* C, defined by f* *(z) = f(z), is also analytic.

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